Parallel postulate

The keynote for this text — the parallel postulate, called the Euclid's fifth postulate:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

How to define a straight line parallel to the other straight line? When the two straight lines are parallel? A trivial thing, right? Not completely. Mathematicians have been thinking about this problem for over 2,000 years. It is basically nothing new, that the most elementary things are the most difficult to define in a clear way. So it was this time. The problem was whether the parallel postulate is result from the previous postulates put forward by Euclid (about 300 BC) and extended by Archimedes, or is it an independent postulate? If the first case occurs, the parallel postulate can be omitted without harming the theory. If there is a second case — it has to be clearly defined. In both cases, clear proof should be provided. The solution to this problem appeared in the nineteenth century, however, in the mathematical world from the seventeenth, there was a passionate discussion.

There are moments characterized by such a huge attention in a field, that the same thought is formulated by several people simultaneously and independently of each other. Newton and Leibniz are considered fathers of differential and integral calculus (small digression: this is a big mistake. Mathematicians have long been interested in the problem of finding a tangent to a given curve - the basic task of differential calculus and finding a field covered by a given curve - the basic task of integral calculus. Newton and Leibniz found a connection between these two issues and introduced a uniform method to mathematics). Let's move on. Fermat and Descartes independently discovered analytical geometry. Dimension theory was formulated by Menger and Urysohn.

Non-Euclidean geometry

Euclid's critics began to appear strongly in the 17th and 18th centuries: Italian mathematician G. Saccheri, Swiss mathematician J. H. Lambert and Frenchman A. M. Legendre. Their contribution consisted mainly in the confrontation with tradition of Euclidean geometry, they did not formulate a new geometry. They consistently developed a geometry called absolute geometry, Euclid's geometry, devoid of the postulate of parallelism. Mathematics had to wait until the nineteenth century for a brave step that formulated the Bolivian-Lobaczewski's non-Euclidean geometry — from the names of the creators. The Hungarian mathematician Janos Bolyai (1802 — 1860) and the Russian mathematician Mikołaj Iwanowicz Łobaczewski (1793 — 1856) decided that they would formulate a geometry in which the postulate of parallelism is denied. It is also worth mentioning here that the German mathematician Carl Friedrich Gauss (1777 — 1855) came to these results much earlier, but he never published them for fear of criticizing ideas so different from those commonly accepted. Gauss lost the priority of discovering non-Euclidean geometry for Łobaczewskiego and Bolyai. Gauss's misgivings, however, were right. The work of Russian and Hungarian mathematicians was widely recognized as scientific pathologies and even referred to as satire directed against mathematics. It did not last long, however — as often happens — appreciation in the scientific world came after the death of the creators. Felix Klein in 1871 formulated the basic idea of ​​the noncontradiction of Bolyai-Łobaczewski's geometry, in 1899 David Hilbert carried out a full proof of the consistency of Euclid's geometry, in 1903 in a similar manner proves the noncontradiction of Bolyai-Lobaczewski's geometry. From this point, both geometries are equally correct in terms of logic.

Gauss's experiment

Bolyai-Łobaczewski's geometry implies a number of new, seemingly absurd, claims — for example, the sum of internal angles in a triangle can be less than 180 degrees. Carl Friedrich Gauss decided to check it out.

He went to the mountains with measuring instruments. He measured the angles in the triangle, which consists of three distant mountain peaks. The result was unambiguous. The sum of the internal angles came almost exactly 180 degrees (within the limits of the error of the instruments). So did this experiment again bury the non-Euclidean geometry that began to germinate? Nothing similar. The Euclidian geometry is in a sense the borderline case of Bolyai-Lobaczewski's geometry. It is obvious that the irrationality in mathematics implies huge limits e.g. in metrology. It is clear that the circumstances in which we do measurements are not perfect. Thus, a problem arises, for which we have no influence — we can not distinguish between a borderline case and its approximation. Even in very close proximity.

For small distances (e.g. several-dozen kilometers) deviations in the hyperbolic geometry are so small that they are within the limits of measurement error. So Gaussian experience have been not revealing. For small figures, both geometries give the same results. However, if we did experiments on relatively large distances (several dozen-several hundred million kilometers — for example to describe the universe), it would turn out that Euclid's geometry fails. This whole story is analogous to the classical mechanics of Newton and the relativistic mechanics of Einstein. Both approaches are convergent for small distances and speeds. However, if the values grow to huge numbers ​​— we have significant differences.

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Graphics and photos:
[1] generated in Texmaker
[2]
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References:

1. Borsuk K., W. Szmielew — Podstawy geometrii, PWN, Warszawa 1970.
2. Courant R., H. Robbins — Co to jest matematyka, PWN, Warszawa 1967.